What is the use of the $\epsilon$ term in this back-propagation equation?
The change in nomenclature from what you expected is Lisa Meeden's choice, for unknown reasons. Those with whom she published in the past used $\epsilon$ to represent error, the result of a loss function. Why she did not use $\alpha$ may be because the letter $a$ was used elsewhere in the formula, but, if that was the reasons, that wasn't a great one.

Changes in nomenclature between articles is common, for instance between Carnegie Mellon literature and Cal Tech. Robotics control systems people use different nomenclature from those who write game competition systems. The online mappings between choice of Greek letters and their meaning cannot be relied upon. This is not particular to AI. I've seen it in science and engineering literature, even between text books in courses that are part of the same university curriculum.